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Theorem lvoli3 33233
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
lvoli3.l  |-  .<_  =  ( le `  K )
lvoli3.j  |-  .\/  =  ( join `  K )
lvoli3.a  |-  A  =  ( Atoms `  K )
lvoli3.p  |-  P  =  ( LPlanes `  K )
lvoli3.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvoli3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )

Proof of Theorem lvoli3
Dummy variables  y 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 992 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  P
)
2 simpl3 993 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  A
)
3 simpr 461 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  -.  Q  .<_  X )
4 eqidd 2444 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  =  ( X 
.\/  Q ) )
5 breq2 4308 . . . . . 6  |-  ( y  =  X  ->  (
r  .<_  y  <->  r  .<_  X ) )
65notbid 294 . . . . 5  |-  ( y  =  X  ->  ( -.  r  .<_  y  <->  -.  r  .<_  X ) )
7 oveq1 6110 . . . . . 6  |-  ( y  =  X  ->  (
y  .\/  r )  =  ( X  .\/  r ) )
87eqeq2d 2454 . . . . 5  |-  ( y  =  X  ->  (
( X  .\/  Q
)  =  ( y 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  r ) ) )
96, 8anbi12d 710 . . . 4  |-  ( y  =  X  ->  (
( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) )  <-> 
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) ) ) )
10 breq1 4307 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  X  <->  Q  .<_  X ) )
1110notbid 294 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  X  <->  -.  Q  .<_  X ) )
12 oveq2 6111 . . . . . 6  |-  ( r  =  Q  ->  ( X  .\/  r )  =  ( X  .\/  Q
) )
1312eqeq2d 2454 . . . . 5  |-  ( r  =  Q  ->  (
( X  .\/  Q
)  =  ( X 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  Q ) ) )
1411, 13anbi12d 710 . . . 4  |-  ( r  =  Q  ->  (
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) )  <-> 
( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) ) )
159, 14rspc2ev 3093 . . 3  |-  ( ( X  e.  P  /\  Q  e.  A  /\  ( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) )
161, 2, 3, 4, 15syl112anc 1222 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) )
17 simpl1 991 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  HL )
18 hllat 33020 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1917, 18syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  Lat )
20 eqid 2443 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
21 lvoli3.p . . . . . 6  |-  P  =  ( LPlanes `  K )
2220, 21lplnbase 33190 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
231, 22syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  (
Base `  K )
)
24 lvoli3.a . . . . . 6  |-  A  =  ( Atoms `  K )
2520, 24atbase 32946 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
262, 25syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  (
Base `  K )
)
27 lvoli3.j . . . . 5  |-  .\/  =  ( join `  K )
2820, 27latjcl 15233 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( X  .\/  Q )  e.  ( Base `  K
) )
2919, 23, 26, 28syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  ( Base `  K ) )
30 lvoli3.l . . . 4  |-  .<_  =  ( le `  K )
31 lvoli3.v . . . 4  |-  V  =  ( LVols `  K )
3220, 30, 27, 24, 21, 31islvol3 33232 . . 3  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  ( Base `  K
) )  ->  (
( X  .\/  Q
)  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) ) )
3317, 29, 32syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( ( X 
.\/  Q )  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) ) )
3416, 33mpbird 232 1  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2728   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   Basecbs 14186   lecple 14257   joincjn 15126   Latclat 15227   Atomscatm 32920   HLchlt 33007   LPlanesclpl 33148   LVolsclvol 33149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-lat 15228  df-clat 15290  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-lplanes 33155  df-lvols 33156
This theorem is referenced by:  dalem9  33328  dalem39  33367
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